*> \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
* LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBQ, JOBU, JOBV
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
* $ V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGGSVD computes the generalized singular value decomposition (GSVD)
*> of an M-by-N real matrix A and P-by-N real matrix B:
*>
*> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R )
*>
*> where U, V and Q are orthogonal matrices.
*> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
*> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
*> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
*> following structures, respectively:
*>
*> If M-K-L >= 0,
*>
*> K L
*> D1 = K ( I 0 )
*> L ( 0 C )
*> M-K-L ( 0 0 )
*>
*> K L
*> D2 = L ( 0 S )
*> P-L ( 0 0 )
*>
*> N-K-L K L
*> ( 0 R ) = K ( 0 R11 R12 )
*> L ( 0 0 R22 )
*>
*> where
*>
*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*> S = diag( BETA(K+1), ... , BETA(K+L) ),
*> C**2 + S**2 = I.
*>
*> R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*> K M-K K+L-M
*> D1 = K ( I 0 0 )
*> M-K ( 0 C 0 )
*>
*> K M-K K+L-M
*> D2 = M-K ( 0 S 0 )
*> K+L-M ( 0 0 I )
*> P-L ( 0 0 0 )
*>
*> N-K-L K M-K K+L-M
*> ( 0 R ) = K ( 0 R11 R12 R13 )
*> M-K ( 0 0 R22 R23 )
*> K+L-M ( 0 0 0 R33 )
*>
*> where
*>
*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*> S = diag( BETA(K+1), ... , BETA(M) ),
*> C**2 + S**2 = I.
*>
*> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
*> ( 0 R22 R23 )
*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The routine computes C, S, R, and optionally the orthogonal
*> transformation matrices U, V and Q.
*>
*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
*> A and B implicitly gives the SVD of A*inv(B):
*> A*inv(B) = U*(D1*inv(D2))*V**T.
*> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is
*> also equal to the CS decomposition of A and B. Furthermore, the GSVD
*> can be used to derive the solution of the eigenvalue problem:
*> A**T*A x = lambda* B**T*B x.
*> In some literature, the GSVD of A and B is presented in the form
*> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 )
*> where U and V are orthogonal and X is nonsingular, D1 and D2 are
*> ``diagonal''. The former GSVD form can be converted to the latter
*> form by taking the nonsingular matrix X as
*>
*> X = Q*( I 0 )
*> ( 0 inv(R) ).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'U': Orthogonal matrix U is computed;
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'V': Orthogonal matrix V is computed;
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBQ
*> \verbatim
*> JOBQ is CHARACTER*1
*> = 'Q': Orthogonal matrix Q is computed;
*> = 'N': Q is not computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> \endverbatim
*>
*> \param[out] L
*> \verbatim
*> L is INTEGER
*>
*> On exit, K and L specify the dimension of the subblocks
*> described in Purpose.
*> K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, A contains the triangular matrix R, or part of R.
*> See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, B contains the triangular matrix R if M-K-L < 0.
*> See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*>
*> On exit, ALPHA and BETA contain the generalized singular
*> value pairs of A and B;
*> ALPHA(1:K) = 1,
*> BETA(1:K) = 0,
*> and if M-K-L >= 0,
*> ALPHA(K+1:K+L) = C,
*> BETA(K+1:K+L) = S,
*> or if M-K-L < 0,
*> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
*> BETA(K+1:M) =S, BETA(M+1:K+L) =1
*> and
*> ALPHA(K+L+1:N) = 0
*> BETA(K+L+1:N) = 0
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,M)
*> If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,M) if
*> JOBU = 'U'; LDU >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,P)
*> If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
*> If JOBV = 'N', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V. LDV >= max(1,P) if
*> JOBV = 'V'; LDV >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
*> If JOBQ = 'N', Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N) if
*> JOBQ = 'Q'; LDQ >= 1 otherwise.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array,
*> dimension (max(3*N,M,P)+N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> On exit, IWORK stores the sorting information. More
*> precisely, the following loop will sort ALPHA
*> for I = K+1, min(M,K+L)
*> swap ALPHA(I) and ALPHA(IWORK(I))
*> endfor
*> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, the Jacobi-type procedure failed to
*> converge. For further details, see subroutine DTGSJA.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLA DOUBLE PRECISION
*> TOLB DOUBLE PRECISION
*> TOLA and TOLB are the thresholds to determine the effective
*> rank of (A',B')**T. Generally, they are set to
*> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
*> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
*> The size of TOLA and TOLB may affect the size of backward
*> errors of the decomposition.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERsing
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
$ IWORK, INFO )
*
* -- LAPACK driver routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
$ V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL WANTQ, WANTU, WANTV
INTEGER I, IBND, ISUB, J, NCYCLE
DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, DLAMCH, DLANGE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
WANTU = LSAME( JOBU, 'U' )
WANTV = LSAME( JOBV, 'V' )
WANTQ = LSAME( JOBQ, 'Q' )
*
INFO = 0
IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( P.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -16
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -18
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGGSVD', -INFO )
RETURN
END IF
*
* Compute the Frobenius norm of matrices A and B
*
ANORM = DLANGE( '1', M, N, A, LDA, WORK )
BNORM = DLANGE( '1', P, N, B, LDB, WORK )
*
* Get machine precision and set up threshold for determining
* the effective numerical rank of the matrices A and B.
*
ULP = DLAMCH( 'Precision' )
UNFL = DLAMCH( 'Safe Minimum' )
TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP
TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP
*
* Preprocessing
*
CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA,
$ TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK,
$ WORK( N+1 ), INFO )
*
* Compute the GSVD of two upper "triangular" matrices
*
CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
$ TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
$ WORK, NCYCLE, INFO )
*
* Sort the singular values and store the pivot indices in IWORK
* Copy ALPHA to WORK, then sort ALPHA in WORK
*
CALL DCOPY( N, ALPHA, 1, WORK, 1 )
IBND = MIN( L, M-K )
DO 20 I = 1, IBND
*
* Scan for largest ALPHA(K+I)
*
ISUB = I
SMAX = WORK( K+I )
DO 10 J = I + 1, IBND
TEMP = WORK( K+J )
IF( TEMP.GT.SMAX ) THEN
ISUB = J
SMAX = TEMP
END IF
10 CONTINUE
IF( ISUB.NE.I ) THEN
WORK( K+ISUB ) = WORK( K+I )
WORK( K+I ) = SMAX
IWORK( K+I ) = K + ISUB
ELSE
IWORK( K+I ) = K + I
END IF
20 CONTINUE
*
RETURN
*
* End of DGGSVD
*
END